SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 1948-1975, January 2021.
The estimation of the probability of rare events is an important task in reliability and risk assessment. We consider failure events that are expressed in terms of a limit-state function, which depends on the solution of a partial differential equation (PDE). In many applications, the PDE cannot be solved analytically. We can only evaluate an approximation of the exact PDE solution. Therefore, the probability of rare events is estimated with respect to an approximation of the limit-state function. This leads to an approximation error in the estimate of the probability of rare events. Indeed, we prove an error bound for the approximation error of the probability of failure, which behaves like the discretization accuracy of the PDE multiplied by an approximation of the probability of failure, the first-order reliability method (FORM) estimate. This bound requires convexity of the failure domain. For nonconvex failure domains, we prove an error bound for the relative error of the FORM estimate. Hence, we derive a relationship between the required accuracy of the probability of rare events estimate and the PDE discretization level. This relationship can be used to guide practicable reliability analyses and, for instance, multilevel methods.

中文翻译：

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具有近似模型的罕见事件概率的误差分析

SIAM 数值分析杂志，第 59 卷，第 4 期，第 1948-1975 页，2021 年 1 月。
估计罕见事件的概率是可靠性和风险评估中的一项重要任务。我们考虑用极限状态函数表示的故障事件，该函数取决于偏微分方程 (PDE) 的解。在许多应用中，无法解析求解 PDE。我们只能评估精确 PDE 解的近似值。因此，罕见事件的概率是根据极限状态函数的近似值来估计的。这会导致对罕见事件概率的估计出现近似误差。事实上，我们证明了故障概率近似误差的误差界限，其行为类似于 PDE 的离散化精度乘以故障概率的近似值，即一阶可靠性方法 (FORM) 估计。这个界限需要故障域的凸性。对于非凸失效域，我们证明了 FORM 估计的相对误差的误差界限。因此，我们推导出稀有事件估计概率所需的准确度与 PDE 离散化水平之间的关系。这种关系可用于指导可行的可靠性分析，例如多级方法。